Question

The area of a right angled triangle is $30c{m}^2$ and the length of its hypotenuse is $13cm$. The length of the shorter side is $5cm$.

• A. True
• B. False

Answer 1

The correct option is A True

Let ABC be the right angle triangle, right angle at B.

Given, area of $△ABC=30{cm}^2$

$\Rightarrow \frac{1}{2}\times BC\times AB=30$

$\Rightarrow BC\times AB=60$ ...(1)

By Pythagoras theorem, we have

${AC}^2=A{B}^2+{BC}^2$

$\Rightarrow A{B}^2+{BC}^2={13}^2=169$

$\Rightarrow (AB+BC{)}^2-2\times AB\times BC=169$

$\Rightarrow (AB+BC{)}^2-2\times 60=169$ ...[using (1)]

$\Rightarrow (AB+BC{)}^2=289$

$\Rightarrow AB+BC=17$ ...(2)

$\Rightarrow AB+\frac{60}{AB}=17$ ....[using (1)]

$\Rightarrow A{B}^2-17AB+60=0$

$\Rightarrow A{B}^2-12AB-5AB+60=0$

$\Rightarrow AB(AB-12)-5(AB-12)=0$

$\Rightarrow (AB-12)(AB-5)=0$

$\Rightarrow AB=12cm$ or $5cm$

If $AB=12cm$, then $BC=5cm$

If $AB=5cm$, then $BC=12cm$ ....[using (2)]

Hence, the shorter leg of the triangle is $5cm$ long.